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A dispersion relation specifies which angular frequencies and wavevectors a linear wave model admits. Fix the plane wave conventionHere $a$ is a constant complex amplitude, $\mathbf{k}$ is the wavevector, and $\omega$ is the angular frequency. Substitution into a constant-coefficient linear partial differential equation replaces ${\partial }_t$ by $-i\omega$ and $\nabla$ by $i\mathbf{k}$, turning the wave problem into an algebraic condition. A system can have several roots; $\omega =\omega (\mathbf{k})$ means one chosen branch, rather than a promise that the full dispersion relation is single-valued. For real $\mathbf{k}$, $\mathrm{Im}\omega >0$ means temporal growth in this convention, while $\mathrm{Im}\omega <0$ means damping. The real-branch discussion of phase velocity and group velocity below assumes negligible loss.

For the one-dimensional wave equation $u_{tt}=c^2{u}_{xx}$, the dispersion relation is ${\omega }^2=c^2{k}^2$. The branch $\omega =ck$ carries right-moving travelling waves for $c>0$; both its phase velocity and group velocity are $c$. One may instead use the positive-angular frequency branch $\omega =c|k|$, with right-moving components at $k>0$ and left-moving components at $k<0$. A real wave pairs each $(k,\omega )$ with $(-k,-\omega )$; that conjugate pair represents the same travelling pattern. Branch and sign choices therefore matter when interpreting a dispersion diagram.

A particularly tangible dispersive example is a small-amplitude surface gravity wave on still water of uniform depth $h$. Neglect surface tension and viscosity, assume incompressible flow and irrotational flow, and impose a flat impermeable bottom. With $k>0$ and positive angular frequency,Here $g$ is gravitational acceleration and $\tanh$ is the hyperbolic tangent. Deep water means $kh\gg 1$: $\omega \simeq \sqrt{gk}$ and longer wavelengths have larger phase speed. Shallow water means $kh\ll 1$: $\omega \simeq \sqrt{gh}k$, an approximately nondispersive wave. Thus depth relative to wavelength, rather than depth alone, determines whether crests and wave packets travel together.^[ref]

The dispersion diagram below also includes the ideal deep-water capillary wave limit $\omega =\sqrt{\gamma /\rho }{k}^{3/2}$, where $\gamma$ is surface tension and $\rho$ is mass density. This limit requires the surface tension restoring term to dominate gravity; it reverses the trend of the surface gravity wave, giving larger phase speed at shorter wavelength. The combined gravity-capillary wave has ${\omega }^2=gk+(\gamma /\rho )k^3$.^[ref]

The curves use dimensionless normalizations chosen separately for each model; they are analytical diagrams, not experimental data. Their beauty is that a single dispersion relation encodes several observable behaviours: its value gives the angular frequency, its origin-to-point slope gives the one-dimensional phase velocity, its tangent slope gives the group velocity, and variation of that tangent slope predicts distortion of a wave packet. A straight line through the origin describes a nondispersive wave; an affine branch $\omega =ck+b$ with $b\ne 0$ has constant group velocity but phase velocity $c+b/k$, so those properties must be distinguished.

For a plane wave with amplitude $\alpha$, wavenumber $k>0$ and angular frequency $\omega >0$, the dispersion relation $\omega =\omega (k)$ specifies which frequencies the governing equations permit. A point of fixed wave phase obeys $kx-\omega t=\text{constant}$, so the phase speed isIt is the speed of individual crests. For a narrow wave packet, two nearby plane waves with differences $\Delta k$ and $\Delta \omega$ have a slowly varying wave envelope depending on $\Delta kx-\Delta \omega t$. Taking the limit gives the group velocityThis is the envelope's propagation speed and, in the present linear lossless wave model, the energy transport speed for a narrow packet.

A dispersive wave has phase speed dependent on wavenumber, so different components can change their relative phases and alter the packet's shape. A nondispersive wave on the chosen propagation branch has constant phase speed over the relevant band:Indeed $d{c}_p/dk=(c_g-c_p)/k$, so equality throughout a band is equivalent to constant phase speed there. Equality at an isolated wavenumber does not make the entire wave family nondispersive.

Define the dimensionless parameterThe stipulated hierarchy $kh\ll \delta \ll 1$ implies both $kh\ll 1$ andThus the finite-depth correction uses $\tanh (kh)\sim kh$, while the restoring effect of bending stiffness is small compared with gravity. The dispersion relation becomesHence, to leading order, these surface gravity waves are nondispersive waves withFor clarity, the first corrections to the phase speed areThey exhibit the weak residual wave dispersion at finite wavenumber. The claimed nondispersive behavior is the asymptotic leading-order result, not exact equality for the complete relation.

A nondispersive wave has a linear dispersion relation $\omega =vk$ over the relevant range, so every Fourier transform component has the same phase velocity $v_p=\omega /k=v$ and group velocity $v_g=d\omega /dk=v$. A wave packet then translates without spreading in an ideal homogeneous lossless medium.

For wave dispersion, $v_p$ depends on frequency or wavenumber. Different components accumulate different phases, and in general the group velocity differs from the phase velocity; variation of group velocity across the spectrum produces spreading of a wave packet. The membrane material can be nondispersive while confinement in a membrane waveguide introduces wave dispersion along its length.

The phase velocity is the motion of a constant-wave phase surface. For a one-dimensional plane wave $u=a\cos (kx-\omega t+{\varphi }_0)$ with real $k\ne 0$ and real $\omega$, fixing $kx-\omega t+{\varphi }_0$ givesEach wave crest of this monochromatic travelling wave advances at that signed phase velocity. Its magnitude is the phase speed $|c_p|=\lambda /T$, where $\lambda =2\pi /|k|$ is the wavelength and $T=2\pi /|\omega |$ is the temporal period when $\omega \ne 0$. Choosing $\omega >0$ makes positive $k$ mean rightward phase velocity and negative $k$ mean leftward phase velocity. Reversing both signs leaves the real plane wave and its phase velocity unchanged. At $k=0$ there are no spatial crests to track, so $\omega /k$ is undefined; a finite limit as $k\to 0$ is a limiting phase speed, not the speed of a spatially uniform oscillation.

In several dimensions, a plane wave has wave phase $\mathbf{k}\cdot \mathbf{x}-\omega t+{\varphi }_0$. Constant-wave phase planes obey $\mathbf{k}\cdot \mathbf{v}=\omega$. Their normal phase velocity isTangential motion within a plane does not move that wavefront, which is why the normal direction fixes the vector convention. In an anisotropic medium the group velocity need not point along this wavevector. Likewise the horizontal phase velocity of a tilted wavefront is its crossing speed along a horizontal line, rather than a component of the normal phase velocity.

For the nondispersive wave branch $\omega =ck$, a wave crest and a wave envelope both move at $c$. For deep-water surface gravity waves, $\omega =\sqrt{gk}$ gives $c_p=\sqrt{g/k}$ and group velocity $c_g=c_p/2$. With the illustrative choices $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ and $\lambda =1\mathrm{m}$, the model predicts $c_p\simeq 1.25\mathrm{m}{\mathrm{s}}^{-1}$ and $c_g\simeq 0.625\mathrm{m}{\mathrm{s}}^{-1}$. These are calculated values for negligible surface tension and sufficiently large $kh$, not measurements. A wave crest moves forward through the wave packet, so timing a crest does not time the packet's journey.

There is a direct laboratory meaning to this distinction. MIT's 2005 surface-wave lab uses a long water tank, a controllable hydraulic paddle, and an absorbing beach. Resistance probes convert submerged length into voltage; calibration converts their readouts into free surface elevation versus time. Two probes a known distance apart, backed by a ruler and stopwatch, let students time a wave crest and separately time a feature of the wave packet. The lab records depth and varies frequency and wave amplitude.^[ref]

For a clean measurement of phase velocity, fit the periodic portion of both records or follow one identifiable wave crest. A phase difference is only known modulo $2\pi$: probe spacing below half the shortest relevant wavelength, or continuity while sweeping frequency, avoids assigning the wrong whole-cycle delay. Then compare the timed phase velocity with $\lambda /T$. Tracking pepper grains instead measures fluid-particle motion, which has a different meaning from either phase velocity or group velocity.

The Klein-Gordon packet velocities example makes the distinction especially sharp: its phase velocity exceeds $c$, yet its narrow wave envelope moves below $c$. A carrier wave labels successive oscillations; its speed alone does not establish information velocity. The front velocity depends on the full causal model, including angular frequencies far outside the carrier's narrow range.